If the standard deviation of the numbers 2, 3, a and 11 is 3.5

If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?

  1. 3a2 – 32a + 84 = 0
  2. 3a2 – 34a + 91 = 0
  3. 3a2 – 23a + 44 = 0
  4. 3a2 – 26a + 55 = 0

Solution

Numbers are 2, 3, a and 11

N = 4

standard Deviation, σ = 3.5

Mean of numbers, μ = (2+3+11+a)/4 = (16+a)/4

\( \sigma^2 = \dfrac{1}{N} \sum (x_i - \mu)^2 \)

3.5 × 3.5 × 4 × 16 = (8+a)2 + (4+a)2 + (16-3a)2 + (a-28)2

= 64 + a2 + 16a + 16 + a2 + 8a + 256 + 9a2 - 96a + a2 + 784  - 56a

= 12a2 - 128a + 1120

196 = 3a2 - 32a + 280

3a2 + 16a + 84 = 0

The correct answer is A.